US9722691B2 - Data detection method and data detector for signals transmitted over a communication channel with inter-symbol interference - Google Patents
Data detection method and data detector for signals transmitted over a communication channel with inter-symbol interference Download PDFInfo
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- US9722691B2 US9722691B2 US14/650,655 US201214650655A US9722691B2 US 9722691 B2 US9722691 B2 US 9722691B2 US 201214650655 A US201214650655 A US 201214650655A US 9722691 B2 US9722691 B2 US 9722691B2
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- H—ELECTRICITY
- H04—ELECTRIC COMMUNICATION TECHNIQUE
- H04B—TRANSMISSION
- H04B7/00—Radio transmission systems, i.e. using radiation field
- H04B7/14—Relay systems
- H04B7/15—Active relay systems
- H04B7/185—Space-based or airborne stations; Stations for satellite systems
- H04B7/1851—Systems using a satellite or space-based relay
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- H—ELECTRICITY
- H04—ELECTRIC COMMUNICATION TECHNIQUE
- H04B—TRANSMISSION
- H04B1/00—Details of transmission systems, not covered by a single one of groups H04B3/00 - H04B13/00; Details of transmission systems not characterised by the medium used for transmission
- H04B1/06—Receivers
- H04B1/10—Means associated with receiver for limiting or suppressing noise or interference
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- H—ELECTRICITY
- H04—ELECTRIC COMMUNICATION TECHNIQUE
- H04B—TRANSMISSION
- H04B1/00—Details of transmission systems, not covered by a single one of groups H04B3/00 - H04B13/00; Details of transmission systems not characterised by the medium used for transmission
- H04B1/69—Spread spectrum techniques
- H04B1/707—Spread spectrum techniques using direct sequence modulation
- H04B1/7097—Interference-related aspects
- H04B1/7103—Interference-related aspects the interference being multiple access interference
- H04B1/7105—Joint detection techniques, e.g. linear detectors
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- H—ELECTRICITY
- H04—ELECTRIC COMMUNICATION TECHNIQUE
- H04L—TRANSMISSION OF DIGITAL INFORMATION, e.g. TELEGRAPHIC COMMUNICATION
- H04L25/00—Baseband systems
- H04L25/02—Details ; arrangements for supplying electrical power along data transmission lines
- H04L25/03—Shaping networks in transmitter or receiver, e.g. adaptive shaping networks
- H04L25/03006—Arrangements for removing intersymbol interference
- H04L25/03171—Arrangements involving maximum a posteriori probability [MAP] detection
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- H—ELECTRICITY
- H04—ELECTRIC COMMUNICATION TECHNIQUE
- H04L—TRANSMISSION OF DIGITAL INFORMATION, e.g. TELEGRAPHIC COMMUNICATION
- H04L25/00—Baseband systems
- H04L25/02—Details ; arrangements for supplying electrical power along data transmission lines
- H04L25/03—Shaping networks in transmitter or receiver, e.g. adaptive shaping networks
- H04L25/03891—Spatial equalizers
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- H—ELECTRICITY
- H04—ELECTRIC COMMUNICATION TECHNIQUE
- H04W—WIRELESS COMMUNICATION NETWORKS
- H04W84/00—Network topologies
- H04W84/02—Hierarchically pre-organised networks, e.g. paging networks, cellular networks, WLAN [Wireless Local Area Network] or WLL [Wireless Local Loop]
- H04W84/04—Large scale networks; Deep hierarchical networks
- H04W84/06—Airborne or Satellite Networks
Definitions
- the invention relates to a method of detecting data transmitted over a communication channel with inter-symbol interference, and to a data detector for carrying out such a method.
- the invention also relates to a method of transmitting data over a communication channel which is optimal when, at the receiving end of the channel, data detection is performed using said method.
- the invention applies in particular, albeit not exclusively, to non-linear channels, and more particularly to satellite communication systems.
- orthogonal signalling that ensures absence of intersymbol interference (ISI)
- ISI intersymbol interference
- DVB-S2 2nd-generation satellite digital video broadcasting
- RRC square-root raised-cosine
- the frequency spacing between carriers in multi-carrier transmission can be reduced (“Frequency packing”) to increase spectral efficiency while introducing some inter-carrier interference (ICI).
- Frequency packing to increase spectral efficiency while introducing some inter-carrier interference (ICI).
- Paper [7] teaches maximizing the achievable spectral efficiency (ASE) for an additive white Gaussian noise (AWGN) channel, with single or multiple carrier transmission. Detection is performed using a suboptimal symbol-by-symbol detector.
- Reference [8] describes a more sophisticated detection algorithm with constrained complexity which, however, is still sub-optimal and limited to the case of a linear channel.
- turbo detection [15, 16, 17] which are based on the exchange of information between two (or more) soft-input soft-output (SISO) devices that iteratively refine the quality of their outputs.
- SISO soft-input soft-output
- turbo detection can be already considered between the inner detector coping with the channel phase noise and the outer LDPC decoder [18].
- MAP maximum-a posteriori
- the soft interference cancellation (SIC) algorithm proposed in [26, 27] for code-division multiple-access systems can still be applied to the linear ISI scenario.
- its complexity is quadratic in the channel memory when the sliding window approach described in [17, 16] is adopted and cannot extended to the case of a non-linear channel.
- the invention aims at providing a method of detecting data transmitted over a nonlinear channel with ISI and possibly ICI, showing symbol error rate performances near to those of an optimal (ideal) detector, but having much lower complexity.
- the invention also aims at providing a data detector implementing such a method and therefore showing limited complexity and high performances.
- the invention also aims at providing a spectrally efficient method of transmitting data over a nonlinear channel, exploiting the performances of the inventive detection method.
- An object of the present invention is then a data detection method according to claim 1 , comprising the steps of:
- Another object of the invention is a data detector according to claim 15 , comprising:
- Yet another object of the invention is a method of transmitting data over a communication channel according to claim 22 , comprising modulating at least one carrier using a succession of pulses having a same shape and a complex amplitude chosen among a discrete set of allowed values defining a finite-order constellation, the method being characterized in that at least one parameter chosen between a bandwidth of said pulses, their shape, and a temporal spacing between said pulses is chosen so as to maximize spectral efficiency when detection is performed using a method according to the invention.
- FIG. 1 shows a low-pass equivalent model of an exemplary communication system comprising a data detector according to the invention
- FIG. 2 is a block diagram of a satellite transponder used in the satellite channel of said communication system, and introducing nonlinearity;
- FIG. 3 is a block diagram of a data detector according to an embodiment of the invention, exchanging information with a SISO decoder;
- FIG. 4 illustrates a channel shortening filter used in the data detector of FIG. 3 ;
- FIG. 5 is a factor graph illustrating the detection algorithm performed by a detector according to a single-user embodiment of the invention
- FIG. 6 is a factor graph illustrating the detection algorithm performed by a detector according to a multi-user embodiment of the invention.
- FIGS. 7A-7C illustrate the characteristics of a nonlinear satellite transponder
- FIG. 8 is a plot illustrating the technical result of the invention.
- the present invention will be described with reference to a frequency-division-multiplexed satellite communication system, whose block diagram is illustrated on FIG. 1 .
- the stream of bits constituting a message to be transmitted is encoded by an encoder (e.g. a turbo encoder), interleaved (this step being optional), then each block of M bits (M ⁇ 0.1) is mapped to a complex symbol belonging to a complex constellation X.
- a shaping filter converts each symbol to a suitable complex pulse p(t), e.g. a Raised Root Cosine (RRC) pulse. Pulses are then used to modulate a radio-frequency carrier (not shown on the figure, which represents a low-pass equivalent model of the actual system) and amplified.
- RRC Raised Root Cosine
- the received signal corresponding to a user is filtered by a front-end filter and sampled.
- the samples are provided as inputs to a detector/decoder whose structure and operation will be described below.
- each detector/decoder exchanges information with the detectors/decoders for “adjacent users”, i.e. for adjacent carriers.
- x ⁇ ( t ) ⁇ n ⁇ ⁇ l ⁇ x n , l ⁇ p ⁇ ( t - nT ) ⁇ e j2 ⁇ l ⁇ ⁇ F u ⁇ t ( 1 )
- x n,l is the symbol transmitted over the lth channel during the nth symbol interval
- F u the frequency spacing between adjacent channels (in the computation of the spectral efficiency, F u can be used as a measure of the signal bandwidth for the uplink).
- the transmitted symbols ⁇ x n,l ⁇ are independent and uniformly distributed and belong to a given zero-mean M-ary complex constellation X.
- the base pulse p(t) has RRC-shaped spectrum (RRC pulse in the following) with roll-off factor ⁇ .
- RRC pulse in the following
- the technique described here applies to general pulses, not necessarily band-limited or satisfying the Nyquist criterion for the absence of ISI for some given value of the symbol interval.
- the satellite channel comprises a satellite transponder for each carrier occupying the entire transponder bandwidth (single-channel-per-transponder).
- the on-board high power amplifier (HPA) can operate close to saturation and hence with high efficiency, but has a nonlinear response.
- An Input Multiplexer (IMUX) filter selects only one carrier, and different carriers are amplified by different transponders.
- An Output Multiplexer (OMUX) filter reduces then the out-of-band power due to the spectral regrowth after nonlinear amplification (see FIG. 2 ).
- the outputs of different transponders are then multiplexed again and it is assumed that the adjacent users have a frequency separation of F d , possibly different from that in the uplink.
- the useful signal at the user terminal is the sum of independent contributions, one for each user, although these contributions are no more, rigorously, linearly modulated due to the nonlinear transformation of the on-board HPA.
- the symbol interval T is shorter than the value ensuring orthogonal signalling, and therefore ISI is present.
- some ISI is necessarily introduced by the nonlinear response of the transponder and the presence of IMUX and OMUX filters.
- frequency separation F u ,F d can be lower than the channel bandwidth, thus introducing some inter-carrier interference (ICI).
- the inventive data detection method only applies to the “single-channel-per-transponder” scenario, considering two cases: single-user receiver and multiple-user receiver.
- detection can be performed using alternative methods known from the prior art, e.g. [11] for single-user receivers and [12] for multiple-users receivers.
- the single-user receiver, single-channel-per-transponder embodiment of the invention will be considered first, due to its lower complexity as only ISI (both linear and nonlinear) has to be considered. Assuming that all adjacent channels will be removed by the IMUX filter (or their interference neglected), the signal at the transponder input is:
- x ⁇ ( t ) ⁇ k ⁇ x k ⁇ p ⁇ ( t - kT ) ,
- the starting point to derive a low-complexity SISO detection algorithm is the approximate signal model described in [13], which is based on an approximate Volterra-series expansion of the satellite channel. This model will be briefly revised.
- the signal at the HPA output can be expressed as a function of x(t) by using a polynomial expansion in which only odd-order terms appear [14].
- the signal is then filtered through the OMUX filter with input response h o (t).
- the signal s(t) at the transponder output can be expressed using a third order nonlinearity model as
- ⁇ s ⁇ ( t ) ⁇ k ⁇ x k ⁇ [ g ( 1 ) ⁇ ( t - kT ) + ⁇ x k ⁇ 2 ⁇ g ( 3 ) ⁇ ( t - kT ) ] ⁇ ⁇ ⁇
- ⁇ h _ ( 3 ) ⁇ ( t 1 , t 2 ) h ( 3 ) ⁇ ( t 2 , t 1 , t 1 ) + h ( 3 ) ⁇ ( t 1 , t 2 , t 1 ) - I ⁇ ( t 1 - t 2 ) ⁇ h ( 3 ) ⁇ ( t 2 , t 2 , t 2 ) . ( 3 )
- s(t) is given in (3) and w(t) is a zero-mean circularly-symmetric Gaussian process with power spectral density 2N 0 .
- r be a proper set of sufficient statistics, extracted from the received signal, collected in a vector of proper length.
- MAP maximum-a-posteriori
- APPs a-posteriori probabilities
- Q ⁇ L is a design parameter (an arbitrary integer) and Re ⁇ • ⁇ denotes the real component of a complex number.
- design parameter Q determines the trade-off between performance (in terms of symbol error rate) and complexity.
- N 0 value higher than the actual noise variance can be used in equations (10), (11) to improve the performance of the sub-optimal detector.
- Equation (12) The meaning of equation (12)—or of a somehow different expression which could be obtained starting from slightly different assumptions—is that the probability mass function can be expressed by the product of N terms, each associated to a symbol x k and proportional to the product of an indicator function I k ( ⁇ k+1 , ⁇ k , x k ), an a priori probability P k (x k ), a first function F k (x k , ⁇ k ) representing the inter-symbol interference due to the Q symbols preceding said symbol x k and of a second function H i (x k , x k ⁇ i ) representing the inter-symbol interference due to the (L ⁇ Q) previous symbols, i.e. symbols identified by an index “i” taking values comprised between (L+1) and Q.
- r) that represent the outcome of the algorithm are computed by marginalizing the probability mass function, expressed by (12), by applying the well-known sum-product algorithm (SPA) [31].
- the SPA When the length of the cycles is at least six, the SPA is generally expected to provide a good approximation of the exact marginals (see [31] for the general treatment, and [32, 33] for appealing applications). Hence, since the considered FG, irrespectively of the values of L and Q, does not contain any cycle of length lower than six, the SPA can be expected to effectively work.
- the APPs computed by the SPA processor are provided to a SISO decoder, whose outputs are: decoded bits and extrinsic information.
- the extrinsic information is returned to the SPA processor and used as the new a priori probability.
- the algorithm is then iterated e.g. for a predetermined number of iterations or until a suitable stopping condition is fulfilled.
- the SPA processor operates on symbols and the decoder operates on bits, the two elements communicates through a (soft) mapper and a (soft) demapper, as well as an interleaver.
- a dotted box contains the elements composing the detector proper: filter back/sampler FB, channel shortener CS (described below), interference canceller SIC and SPA processor.
- the different elements of the detector and the decoder can be implemented using dedicated hardware, software run on programmable hardware or hybrid hardware/software solutions, as known in the arts of electronics and signal processing.
- Equations 3-12 only consider lowest-order (i.e. third order) nonlinearity of the satellite transponder. However, taking into account higher-order contributions to the nonlinear response of the transponder (more generally: of the channel) does not pose any fundamental problem.
- the signal s(t) at the output of the transponder can be written as:
- the pulse h (5) (t 1 , t 2 , t 3 ) is proportional to the approximate fifth-order Volterra kernel.
- the approximation consists of holding only the terms in the form
- This new signal model can lead to the same factorization (12) of the joint APP of the transmitted sequence and hence to the same FG of FIG. 5 .
- the expressions of the relevant factors in the graph are more involved but the computational load per iteration remains unmodified.
- Detection can be improved without increasing significantly the complexity of the algorithm by using interference cancellation, and preferably soft interference cancellation (SIC), to take into account more than L interfering symbols.
- SIC soft interference cancellation
- r k ( 1 ) ⁇ i ⁇ h i ( 1 ) ⁇ x k - i + ⁇ i ⁇ h i ( 1 , 3 ) ⁇ x k - i ⁇ ⁇ x k - i ⁇ 2 + n k ( 1 )
- r k ( 3 ) ⁇ i ⁇ h - i ( 1 , 3 ) * ⁇ x k - i + ⁇ i ⁇ h i ( 3 ) ⁇ x k - i ⁇ ⁇ x k - i ⁇ 2 + n k ( 3 )
- n k (1) and n k (3) are proper noise samples. It is assumed that, at a given iteration, the equalization algorithm is activated with a set of a priori probabilities, coming from the SISO decoder, equal to ⁇ P k (x k ) ⁇ .
- the algorithm performs SI self-iterations (being SI ⁇ 0 a design parameter) proceeding as follows:
- the extrinsic probabilities fed to the SISO decoder are ⁇ p (SI) (r
- soft interference cancellation is performed by an interference canceller SIC arranged upstream from the detection processor SPA, receiving a priori information from the decoder and extrinsic information from the detector and passing to the detector processed samples r k (1) , r k (3) and updated noise variance values.
- an interference canceller SIC arranged upstream from the detection processor SPA, receiving a priori information from the decoder and extrinsic information from the detector and passing to the detector processed samples r k (1) , r k (3) and updated noise variance values.
- the receiver complexity can be reduced for a given performance or the performance improved for a given complexity by adopting the channel shortening technique described in [34], properly extended here to the channel at hand.
- the channel shortening aims at finding the optimal digital filter (known as “channel shortener”) for the samples at the output of the matched filter, and the optimal coefficients of ISI to be set at detector (known as target response).
- this technique finds the channel shortener and the target response that maximize the achievable information rate of the detector.
- the detector is designed taking into account the target response instead of the actual response of the channel including the channel shortener; in turn, the target response is chosen in order to maximize the information rate.
- the optimization problem is convex with nonlinear constraints.
- the solution can be carried out with standard numerical optimization methods with the limit of complexity due to the problem dimensionality.
- closed-form expressions can be found, under suitable hypothesis.
- the channel shortener and target response expressions are found for the linear channel assuming independent transmitted symbols x k belonging to a constellation with Gaussian distribution.
- simulation results in [34] show that good improvements are achieved also for low-order discrete constellations, like the PSK modulations.
- the framework is generalized here to the nonlinear satellite channel.
- the channel shortener can be in general a two-dimensional digital filter having impulse response ⁇ L i ⁇ , where:
- the target response are the ⁇ tilde over (h) ⁇ i (1) , ⁇ tilde over (h) ⁇ i (1,3) , ⁇ tilde over (h) ⁇ i (3) to be set at the detector.
- H ij ( h j - i ( 1 ) h j - i ( 1 , 3 ) h - ( j - i ) ( 1 , 3 ) * h j - i ( 3 ) )
- 2 ] T and the noise samples n i [n i (1) , n i (3) ] T .
- V is a block matrix such that:
- V d E ⁇ x i x i H ⁇ and does not depend on the time i.
- I R log det ( ⁇ tilde over (H) ⁇ V+I ) ⁇ Tr ( LF H [FVF H +2 N 0 I]FL H [ ⁇ tilde over (H) ⁇ +V ⁇ 1 ] ⁇ 1 )
- L, ⁇ tilde over (H) ⁇ are block Toeplitz matrix representing the channel shortener and the target response respectively.
- the optimal channel shortener ⁇ L i ⁇ , and the target response h i (1) , h i (1,3) , h i (3) to be set in (10) and (11), are thus found in closed form maximizing the achievable information rate by means of mathematical analysis. Taking the derivative of I R with respect to L, ⁇ tilde over (H) ⁇ and setting it to zero, we can find the optimal channel shortener and target response.
- L ( ⁇ ) [ ⁇ tilde over ( H ) ⁇ ( ⁇ )+ V d ⁇ 1 ]V d F ( ⁇ ) H [F ( ⁇ ) V d F ( ⁇ ) H +2 N 0 I] ⁇ 1 ( F ( ⁇ ) H ) ⁇ 1
- L( ⁇ ) is the discrete-time Fourier transform (DTFT) of ⁇ L i ⁇ .
- FIG. 4 shows the generalized receiver based on n-order Volterra-series representation, employing CS.
- This sufficient statistic r i is filtered by a k-dimensional filter whose output is used by the detection algorithm.
- the channel shortener (or channel shortening filter) CS is arranged between the filter bank FB and the interference canceller SIC (if present) or the SPA processor.
- the operation of the SPA processor has to take into account the presence of the channel shortener: in equations (10) and (11) impulse responses h i (1) , h i (3) and h i (1,3) are replaced by the target responses ⁇ acute over (h) ⁇ i (1) , ⁇ acute over (h) ⁇ i (1,3) , ⁇ acute over (h) ⁇ i (3) .
- ICI inter-carrier interference
- the approximated received signal can be expressed by
- Equation (16) The meaning of equation (16)—or of a somehow different expression which could be obtained starting from slightly different assumptions—is that the probability mass function can be expressed by the product of N ⁇ U terms, each associated to a symbol x k,l and proportional to the product of an indicator function I k,l (x k,l , ⁇ k,l , ⁇ k+1,l ), an a priori probability P k,l (x k,l ), a first function F k,l (x k,l , ⁇ k,l ) representing the inter-symbol interference within carrier l due to the Q symbols preceding said symbol x k,l , of a second function
- ⁇ i Q + 1 L ⁇ G k , k - i l , l ⁇ ( x k , l , x k - i , l )
- ⁇ i 1 L ⁇ ⁇ j ⁇ l U ⁇ G k , k - i l , j ⁇ ( x k , l , x k - i , j ) ⁇ ⁇ j > l U ⁇ G k , k l , j ⁇ ( x k , l , x k , j )
- the inventive detector/detecting method it is advantageous to use an optimized transmission scheme wherein the values of T, F u , F d , and the shape of the adopted pulses are chosen in order to maximize the spectral efficiency, thus allowing for ISI and ICI. It can be shown that the problem of determining the optimal pulse shape can be reduced to a finite-dimensional optimization problem, with dimension depending on the channel memory; this is rigorously true in the case of a linear channel, and approximately true for a nonlinear channel.
- the spectral efficiency measures the amount of information per unit of time and bandwidth and is given by the ratio between the information rate and the product between the symbol time and the occupied bandwidth.
- the information rate in bits per channel use, can be computed—for a given detection/decoding scheme—by means of the simulation-based method described in [35].
- a further degree of freedom is represented by the bandwidth of the shaping pulse that can be increased, thus increasing the interference for given values of F u and F d .
- This optimal values of T, F u , and F d , and the shape or bandwidth of the pulse p(t) depend on the employed detector—the larger the interference that the receiver can cope with, the larger the spectral efficiency and the lower the values of T, F u , and F d .
- This technique provides increased spectral efficiencies at least for low-order modulation formats.
- dense constellations with shaping [9] it reduces to a scenario similar to that of the Gaussian channel with Gaussian inputs for which orthogonal signalling is optimal (although this is rigorously true for the linear channel and not in the presence of a nonlinear HPA, since shaping increases the peak-to-average power ratio). Improving the ASE without increasing the constellation order is very convenient since the larger the constellation size, the higher the decoding complexity.
- low-order constellations are more robust to channel impairments such as time-varying phase noise and non-linearity.
- time and frequency spacings are chosen as the minimum values ensuring the same performance as in the case of absence of ISI with the optimal receiver.
- FIGS. 7A-C and 8 The technical result of the invention will now be discussed with reference to a specific example illustrated by FIGS. 7A-C and 8 .
- FIGS. 7A and 7B shows the transfer functions of the IMUX and OMUX of a transponder operating in the ku-band
- FIG. 7C shows the nonlinear Single Carrier Transfer Characteristics of the TWTA (Travelling Wave Tube Amplifier) HPA of said transponder. More precisely, the continuous curve (AM/AM) illustrates the relationship between input amplitude and output amplitude (left ordinate scale), while the dotted curve (AM/PM) illustrates the relationship between input amplitude and output phase (right ordinate scale).
- TWTA Travelling Wave Tube Amplifier
- FIG. 8 show a plot of the spectral efficiency ⁇ as a function of C sat /N, where C sat is the saturation power when a non-modulated carrier (“pure tone”) is transmitted and N is the noise power, for two different scenarios:
- the code-frame length is 64,800 bits
- the transmit pulses are RRC with roll-of 20%
- a DVB-S2-LDPC code is used and the rates are:
- the present invention allows an increase of about 20% in spectral efficiency.
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Abstract
Description
-
- a. receiving a signal transmitted over a communication channel, said signal being representative of at least a stream of interfering symbols xk, each representing one or more bits of a transmitted message;
- b. filtering the received signal through at least a filter bank comprising at least a first filter representative of a linear response of said channel and a second filter representative of a non-linear response of said channel, and sampling the filtered signals at the symbol rate, thus obtaining respective sequences of filtered samples rk (1), rk (3); and
- c. jointly computing the a posteriori probabilities of N>1 consecutive symbols xk by applying the sum-product algorithm to a joint probability mass function expressed by the product of N terms, each being associated to a symbol xk and being proportional to the product of an indicator function, an a priori probability, a first function representing the inter-symbol interference due to the Q symbols which precede said symbol xk and of a second function representing the inter-symbol interference due to the (L−Q) symbols immediately preceding said Q symbols which precede symbol k;
-
- L is an integer greater than zero, representing the memory of the nonlinear channel;
- Q is an integer greater than zero and not greater than L, serving as a design parameter;
-
- at least a filter bank comprising at least a first filter representative of a linear response of a communication channel and a second filter representative of a non-linear response of said channel;
- a sampler for sampling the outputs of said filters at a symbol rate of a received signal, thus obtaining respective sequences of filtered samples rk (1),rk (3); and
- a processor for taking said sequences of filtered samples as inputs and realizing simultaneous soft-input-soft-output detection of a packet of N interfering symbols xk by applying the sum-product algorithm to a probability mass function expressed by the product of N terms, each being associated to a symbol xk and being proportional to the product of an indicator function, an a priori probability, a first function representing the inter-symbol interference due to the Q symbols which precede said symbol xk and of a second function representing the inter-symbol interference due to the (L−Q) symbols immediately preceding said Q symbols which precede symbol k.
are to me so-called Volterra kernels of first and third order, respectively. In [13], an approximate Volterra representation is used to simplify the signal expression at the transponder output. The approximation consists of holding only the terms in the form xp|xq|2 among the 3-symbols products xi.xj.xl* which appear in the polynomial expansion (2), and using the approximation |xp|2=E{|xp|2} when p≠q. The signal thus becomes
r(t)=s(t)+w(t) (4)
h i (1)=∫−∞ ∞ g (1)(t)g (1)*(t−iT)dt
h i (3)=∫−∞ ∞ g (3)(t)g (3)*(t−iT)dt
h i (1,3)=∫−∞ ∞ g (3)(t)g (1)*(t−iT)dt
r i (1)=∫−∞ ∞ r(t)g (1)*(t−iT)dt (7)
r i (3)=∫−∞ ∞ r(t)g (3)*(t−iT)dt (8)
P(x,σ| r)∝p(r|x,σ)P(o|x)P(x)=p(r|x)P(σ|x)P(x).
From the state definition it follows that the state in a generic time instant is completely determined by the previous state and the symbol transmitted in the previous interval, and hence the probability P(σ|x) can be factorized as
where Ik(σk+1, σk, xk) is the trellis indicator function, equal 1 if the next state σk+1 is compatible with the current state σk and the symbol xk. By using (5) and (9) and introducing the functions
the probability mass function P(x, σ|r) can be factorized as
r i=(r i (1) ,r i (3))T.
r=Hx+n
E{xx H }=V
I R=log det({tilde over (H)}V+I)−Tr(LF H [FVF H+2N 0 I]FL H [{tilde over (H)}+V −1]−1)
where F=Chol(H) is the Cholesky factorization of H, and L, {tilde over (H)} are block Toeplitz matrix representing the channel shortener and the target response respectively.
L(ω)=[{tilde over (H)}(ω)+V d −1 ]V d F(ω)H [F(ω)V d F(ω)H+2N 0 I] −1(F(ω)H)−1
where L(ω) is the discrete-time Fourier transform (DTFT) of {Li}.
B(ω)=V d −V a F(ω)H [F(ω)V d F(ω)H+2N 0 I] −1 F(ω)V d,
b=[b 1 , . . . ,b L].
B ij =b i−j
and:
c=b 0 −b H B −1 b.
{tilde over (H)}(ω)=U(ω)H U(ω)−V d
A sufficient statistic for signal (13) can be obtained through a bank of filters matched to the pulses g(1)(t) and g(3)(t) and properly centred at each frequency lFd. At discrete time k, the output of the two filters for user m can be expressed as:
and (nk,m (1), nk,m (3)) are coloured circularly-symmetric zero-mean Gaussian random variables. Equations (14) and (15) can be rewritten, for each k and m, in a matrix form as
r=Hx+n
where the matrix H is a block matrix composed of sub-matrices of the form
and the vectors x, r, n are obtained by the concatenation of the sub-vectors
x k,l=(x k,l ,x k,l |x k,l|2)T
r k,l=(r k,l (1) ,r k,l (3))T
n k,l=(n k,l (1) ,n k,l (3))T
respectively. By introducing the functions
where σk,l=(xk−1,l, xk−2,l, . . . , xk−Q,l), the following factorization of the APP can be obtained:
-
- The dotted line with circles refers to the application of a single-user detection algorithm according to the invention, with optimal time-packing (TP), channel shortening (CS) without SIC, L=1 and Q=L;
- The dotted line with crosses, to the case of orthogonal signalling and symbol-by-symbol detection.
-
- For the method according to the invention:
- QPSK: 1/3; 1/2; 3/5
- 8PSK: 1/2; 3/5; 2/3
- 16APSK: 2/3; 3/4
- 32APSK: 2/3; 3/4; 5/6; 8/9
- For the method with orthogonal signalling:
- QPSK: 1/2; 3/5; 3/4
- 8PSK: 3/5; 3/4; 8/9
- 16APSK: 3/4; 4/5; 5/6; 8/9
- 32APSK: 3/4; 4/5; 5/6; 8/9; 9/10.
- For the method according to the invention:
- [1] ETSI EN 301 307 Digital Video Broadcasting (DVB); V1.1.2 (2006-06), Second generation framing structure, channel coding and modulation systems for Broadcasting, Interactive Services, News Gathering and other Broadband satellite applications, 2006.
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Claims (23)
h i (1)=∫−∞ ∞ g (1)(t)g (1)*(t−iT)dt;
h i (3)=∫−∞ ∞ g (3)(t)g (3)*(t−iT)dt;
h i (1,3)=∫−∞ ∞ g (3)(t)g (1)*(t−iT)dt;
g (3)(t)=¾γ3
h i (1)=∫−∞ ∞ g (1)(t)g (1)*(t−iT)dt;
h i (3)=∫−∞ ∞ g (3)(t)g (3)*(t−iT)dt;
h i (1,3)=∫−∞ ∞ g (3)(t)g (1)*(t−iT)dt;
g (3)(t)=¾γ3
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US11451419B2 (en) | 2019-03-15 | 2022-09-20 | The Research Foundation for the State University | Integrating volterra series model and deep neural networks to equalize nonlinear power amplifiers |
US11855813B2 (en) | 2019-03-15 | 2023-12-26 | The Research Foundation For Suny | Integrating volterra series model and deep neural networks to equalize nonlinear power amplifiers |
US12273221B2 (en) | 2019-03-15 | 2025-04-08 | The Research Foundation For The State University Of New York | Integrating Volterra series model and deep neural networks to equalize nonlinear power amplifiers |
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